Equipartition of Energy

The theorem of equipartition of energy states that molecules in thermal equilibrium have the same average energy associated with each independent degree of freedom of their motion and that the energy is

The equipartition result

serves well in the definition of kinetic temperature since that involves just the translational degrees of freedom, but it fails to predict the specific heats of polyatomic gases because the increase in internal energy associated with heating such gases adds energy to rotational and perhaps vibrational degrees of freedom. Each vibrational mode will get kT/2 for kinetic energy and kT/2 for potential energy - equality of kinetic and potential energy is addressed in the virial theorem. Equipartition of energy also has implication for electromagnetic radiation when it is in equilibrium with matter, each mode of radiation having kT of energy in the Rayleigh-Jeans law.

For the translational degrees of freedom only, equipartition can be shown to follow from the Boltzmann distribution.

Thermal Energy

The average translational kinetic energy possessed by free particles given by equipartition of energy is sometimes called the thermal energy per particle. It is useful in making judgements about whether the internal energy possessed by a system of particles will be sufficient to cause other phenomena. It is also useful for comparisons of other types of energy possessed by a particle to that which it possesses simply as a result of its temperature.

(The default mass of 18 amu, the mass of a water molecule, is used in the above calculation if you do not choose a value. The velocity of a water molecule at different temperatures is a good example of the molecular speeds associated with the internal energy of molecules.)

The calculated thermal energy provides a useful comparison for the energies involved in other physical phenomena. For example, in the interaction of radiation with matter it is useful to compare the quantum energy of the photons of the radiation with the thermal energy at the existing temperature. The photon energy associated with the photons in a microwave oven at frequency 2.45 GHz is about 10-5 eV. The average thermal energy at 20°C is 0.04 eV, about 3700 times higher. This suggests that the specific effects of the microwave photons in rotating molecules will be quickly randomized by the overwhelming thermal energy of the surroundings, so that the contribution of the microwaves will be "thermalized" or appear as a contribution to raising the temperature of the material.

Internal Energy for Ideal Gas

Internal energy in general includes both kinetic energy and potential energy associated with the molecular motion. But the potential energy is associated with intermolecular forces and is presumed to be zero in an ideal gas where the only molecular interactions are the perfectly elastic collisions between molecules. Therefore the internal energy of an ideal gas is entirely kinetic energy.

While steam at 100 degrees Celsius is not strictly an ideal gas, the diagram illustrates the fact that the phase change to the gaseous state leaves only the kinetic portion of the internal energy. For
a monoatomic ideal gas this internal energy is given by

If rotation and vibrational kinetic energies are involved (polyatomic molecules) then

Study of the specific heats of gases provides evidence of whether or not rotation and vibration play a significant role in the molecular kinetic energy.